My favourite is Lizzie, like in Pride and Prejudice (though in the book it's spelled Lizzy). But I prefer the full Elizabeth over any of the nicknames.

I think it's fine for you and your husband to call her by different nicknames. One or both could be her special nickname from Mom/Dad, and the other (or another one entirely) the one she goes by in general. Especially once she gets older and prefers a certain one (or something else altogether).